The origin of the Willmore energy can be traced to Siméon-Denis Poisson and Sophie Germain who had engaged in a scientific contest to give a mathematical explanation of what is now known as the Chladni figures. The prize was won in 1816 by Sophie Germain.
She had studied the elasticity properties of membranes and thin plates and her work brought into limelight an entirely new area of applied mathematics, namely, elasticity theory. Her work shows that the bending energy of a thin plate, in its simplest form, can be written (using our notation) as (REF ) (see [1]}). Problems involving elastic surfaces can be studied via the energy (REF ) (see [2]}). Since then, the Willmore energy has been widely studied in differential geometry and applied to some areas of science such as General Relativity [3]}, [4]} (where the Willmore energy is the main contributor to the Hawking mass), Cell Biology [5]} (where the Willmore energy plays a major role in the Helfrich model), String theory [6]}, and Image Processing [7]}. The Willmore energy also relates to the renormalised area functional for a minimal surfaces embedded in hyperbolic 3-manifold [8]}.
Earlier works include those of G. Thomsen who established the Euler-Lagrange equation for the energy (REF ) [9]}, W. Blaschke who observed minimal surfaces are minimizers of the energy (REF ) [10]}, J. Weiner who generalised the work of Thomsen in higher codimension [11]} and T. J. Willmore who explored the lower bounds for the Willmore energy of a closed orientable surface [12]}. Actually, the Willmore energy was named after him.
Of great importance is the Willmore conjecture ([12]}, see also [14]}) which was proved in [15]} by Fernando Codá Marques and André Neves.
